Section 3.4: Work-Energy Theorem

The work-energy theorem states that the net work done on an object by all forces is equal to the change in its kinetic energy:

\( W_\text{net} = \Delta KE = KE_f - KE_i \)

Where \( W_\text{net} \) is the total work done by all forces acting on the object, and \( KE = \frac{1}{2} m v^2 \).

Important Notes:
  • Positive work increases the object's kinetic energy.
  • Negative work decreases kinetic energy.
  • The theorem applies even when forces vary in magnitude or direction.

Example 1

A 5 kg box is pushed along a frictionless surface by a 20 N horizontal force for 4 m. Find the final speed of the box if it starts from rest.

Work done: \( W = F \cdot d = 20 * 4 = 80 \text{ J} \)

Change in KE: \( \Delta KE = W = \frac{1}{2} m v^2 \Rightarrow 80 = 0.5*5*v^2 \Rightarrow v = \sqrt{32} \approx 5.66 \text{ m/s} \)

Example 2

A 2 kg object moving at 3 m/s is acted on by a force that does -8 J of work. Find its final speed.

Initial KE: \( KE_i = 0.5 * 2 * 3^2 = 9 \text{ J} \)

Work done: \( W = -8 \text{ J} \)

Final KE: \( KE_f = KE_i + W = 9 - 8 = 1 \text{ J} \)

Final speed: \( v_f = \sqrt{\frac{2*1}{2}} = 1 \text{ m/s} \)

Practice Problems

  1. A 3 kg block is pushed by 15 N over 2 m on a frictionless surface. Find final speed starting from rest.
  2. A 5 kg mass moving at 4 m/s experiences 20 J of work by a net force. Find final speed.
  3. An object of mass 2 kg slows down due to 10 J of work done by friction. Initial speed = 5 m/s. Find final speed.
  4. A 1 kg mass is pulled with 10 N at 30° over 3 m. Find work done and final speed starting from rest.
  5. A car of mass 800 kg accelerates with net work 6400 J. Initial speed = 10 m/s. Find final speed.
  6. A block of mass 3 kg is pushed with varying force along 5 m. Net work done = 45 J. Find final velocity.
  7. An object initially at rest is accelerated by 12 N along 6 m. Mass = 4 kg. Find final speed.
  8. A 2 kg object moving at 6 m/s has net work -18 J applied. Find final speed.
  9. A box of 10 kg is lifted vertically 2 m. Work done by gravity? KE change if released?
  10. A 0.5 kg block slides down frictionless incline of 1.5 m. Work done by gravity = ? Final speed?
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